where
x∈ℝ3, the
frictional coefficient
α(t)=μ∕(1+t)λ
with
μ>0
and
λ≥0,
ρ̄>0 is a
constant,
ρ0,u0∈C0∞(ℝ3),
(ρ0,u0)≢0,
ρ(0,x)>0,
curlu0≡0, and
ε>0 is sufficiently small. For
0≤λ≤1, we show that there
exists a global
C∞([0,∞)×ℝ3)-smooth
solution
(ρ,u)
by introducing and establishing some uniform time-weighted energy estimates of
(ρ,u), while for
λ>1, in general, the smooth
solution
(ρ,u) blows up in
finite time. Therefore,
λ=1
appears to be the critical value for the global existence of small amplitude smooth
solution
(ρ,u).